A water tank is being filled at a constant rate. After 3 hours of filling, the tank contains 950 gallons...

1. TRANSLATE the problem information

• Given information:
  • At \(\mathrm{t = 3}\) hours: \(\mathrm{w = 950}\) gallons
  • At \(\mathrm{t = 7}\) hours: \(\mathrm{w = 1{,}350}\) gallons
  • Tank fills at constant rate

• What this tells us: We have two coordinate points \(\mathrm{(3, 950)}\) and \(\mathrm{(7, 1350)}\), and since the rate is constant, this is a linear relationship.

2. INFER the solution approach

• Since we need a linear equation \(\mathrm{w = mt + b}\), we must:
  • Find the rate of filling (slope m) using our two points
  • Find the initial amount in tank (y-intercept b) using the rate and one point

• Strategy: Calculate slope first, then substitute back to find initial amount

3. SIMPLIFY to find the rate of filling

• Using slope formula with points \(\mathrm{(3, 950)}\) and \(\mathrm{(7, 1350)}\):

\(\mathrm{Rate = \frac{1350 - 950}{7 - 3}}\)
\(\mathrm{= \frac{400}{4}}\)
\(\mathrm{= 100}\) gallons per hour

4. SIMPLIFY to find initial amount in tank

• Using \(\mathrm{w = 100t + b}\) with point \(\mathrm{(3, 950)}\):

\(\mathrm{950 = 100(3) + b}\)
\(\mathrm{950 = 300 + b}\)
\(\mathrm{b = 650}\) gallons

5. Write final equation and verify

• Equation: \(\mathrm{w = 100t + 650}\)
• Check with second point: \(\mathrm{w = 100(7) + 650 = 1350}\) ✓

Answer: A) \(\mathrm{w = 100t + 650}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly think the y-intercept is the amount at \(\mathrm{t = 3}\) hours (950 gallons) instead of recognizing they need to find the initial amount when \(\mathrm{t = 0}\).

They correctly find the rate as 100 gallons/hour, but then write: \(\mathrm{w = 100t + 950}\)

This leads them to select Choice C (\(\mathrm{w = 100t + 950}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "rate" means and use the total change in gallons (400) as the rate instead of dividing by the time change.

They write: \(\mathrm{w = 400t + b}\), then when solving for b get confused and often guess at the y-intercept value.

This may lead them to select Choice B (\(\mathrm{w = 400t + 750}\))

The Bottom Line:

This problem requires understanding that linear equations describe relationships over time, and the y-intercept represents the starting value (when \(\mathrm{t = 0}\)), not just any given point on the line. Students must distinguish between "rate of change" (slope) and "total change" (difference in values).